Continuum percolation at and above the uniqueness treshold on homogeneous spaces

Details Continuum percolation at and above the uniqueness treshold on homogeneous spaces Continuum percolation at and above the uniqueness treshold on homogeneous spaces

2007-11-02

arxiv.org/abs/0711.0307v3

Mathematics

Probability

16 pages, corrections made

Scientific paper

We consider the Poisson Boolean model of continuum percolation on a homogeneous Riemannian manifold $M$. Let $lambda$ be intensity of the Poisson process in the model and let $lambda_u$ be the infimum of the set of intensities that a.s. produce a unique unbounded component. We show that above $\lambda_u$ there is a.s. a unique unbounded component. We also study what happens at $\lambda_u$ for some spaces. In particular, if $M$ is the product of the hyperbolic disc and the real line, then at $\lambda_u$ there is a.s. not a unique unbounded component. The results are inspired by results for Bernoulli bond percolation on graphs due to Haggstrom, Peres and Schonmann.

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